1. Let Y = [Y1 Y2]T and X = [X1 X2]T be two random vectors such that Y = AX, where A is a 2 × 2 invertible real matrix. Determine fy in terms of fx. %3D

Q: S = Consider the matrix -1 1 1 1 0 [33 3 Find an orthogonal S such that S-¹ AS = D, a diagonal…

A: Given A=0330 and S-1AS=D, a diagoanl matrix

Q: 4. Suppose that xa is an approximate solution to the system of Ax = b. Let A, and I are nonsingular…

A: Sol

Q: Find a 3 x 3 matrix A such that Ax = 9x for all x in R3. A II

A:

Q: [2.25] Show that C= {x : Ax <0}, where A is an m × n matrix, has at most one extreme point, the…

A: 2.25)   show that C=x:Ax≤0 where A is an m×n matrix , has at most one extreme point, the origin.…

Q: 1) Construct the matrix A =(a;;) where 3x2 ) aj = i+j, if i = j,and Li-j, if itj %3D

A:

Q: The mist Consider ann x m matrix A. Find dim(im(A)) + dim(ker(A")), in terms of m and n.

A: Please check step 2 for the solution.!

Q: 1. For the matrix A, let T: R* → R' be defined by T(X) = Af for each i in R'. A is given: [2 1 1 2 A…

A: The given linear transformation is T:ℝ4→ℝ5 is defined as: Tx=Ax for each x∈ℝ4.Where:…

Q: If an m x n matrix U has orthonormal columns, then UTU = I.

A:

Q: Let A and B be two matrices of size 4 x 4 such that det(A) 1. If B is a singular matrix then…

A:

Q: Compute the rank and nullity of the given matrices over the indicated Zp

A:

Q: Show that if A is a normal matrix, then each of the following matrices must also be normal: 1. AH 2.…

A:

Q: T(x, y) = (-x, 4x + 8y, 9x − 3y, 6y – 5x). Find its associated matrix A. A = JUL

A:

Q: Let S = {V1, V2} and T = {w1, w2} be ordered bases for P1, where W1 = 1-1, w2 =t+1. 1 2 If the…

A: Given: S=v1,v2 and T=w1,w2 are the ordered bases for P1, where w1=t-1, w2=t+1 Also, the transition…

Q: 8). Compute the Hessian matrix for functions (a) f(x, y, z) = xy³² – ² – y; (b) f(x, y, z) = xyz;…

A:

Q: 2. If A1(t) and A2(t) are continuous in (a, b), and ¤(t) is a fundamental matrix of dX dX the…

A:

Q: If A is an q x n matrix with orthonormal columns & W be the column space, then AA"y Projwy for all y…

A: Given that A is a q×n matrix with orthonormal columns. Hence there are q rows and n columns in A.…

Q: Which of the following statements are true? Select one or more: Oa. The correlation coefficient…

A: Given :  y1 =e1'X ,..yp=ep'X are principal components . Principal Component analysis is used in…

Q: 5. Compute the condition number K(A) of the following matrix 1 1+€ 1 1 1 A 1 1 1 1+€ What is k(A) as…

A: Conditional number is κ∞(A)=∥A∥∞·∥A-1∥∞.If matrix A is known to us so ∥A∥∞ is easy to find .First…

Q: Consider the 3 x 3 matrix A and the vector & e R³ given below 0 1 0 1 0 1 0 1 0 1 A = i = Prove that…

A:

Q: Given that we have a linear model y =XB+e where y denotes an n x1 vector of dependent variable…

A:

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. If B is a singular matrix then…

A: We have given  det(A) = 3 If B is singular matrix  det(B)  = 0 detBT=det(B) =0 So,  det2A-2BT =0…

Q: Let X(t) be a fundamental matrix for x′ = A(t)x on the interval I. Q. If t0 ∈ I, show that the…

A: Let X(t) be a fundamental matrix for x' = A(t) x t0∈I To show: WE have to show that the solution of…

Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = O \beta…

A:

Q: Generate the matrix A of order n= 100, whose generic element is a; = sin(max(i, j)) The condition…

A:

Q: 11. Let A be an m x n matrix and let x, y e R". Prove that if Ax = 0 and y = ATb for some b e R",…

A:

Q: . A population with two age classes has a Leslie matrix L . If the initial population vector x0,…

A:

Q: **32. Suppose A is an m x n matrix and x e R" satisfies (ATA)x = 0. Prove that Ax = 0. (Hint: What…

A: Here we use property of modular and dot product.

Q: Let A be an nxn matrix. If A-= det A adj A is computed, what should AA be equal to in order to…

A: Using the given matrix equation we need to find the value of AA-1 .

Q: If A is an q x n matrix with orthonormal columns & W be the column space, then AA" y = Projwy for…

A: If A is an invertible qxn matrix with orthonormal columns &amp; W be the column space, then the…

Q: . Let M be the following matrix over Z2: 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 M = Which one of…

A: Given  A 5x4 matrix is given. Nullity = n- rank Nullity = 5 - rank

Q: Let B be an n X n matr ix and let E be an n X n elementary matrix. Then det(EB) = (det E)(det B)

A:

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. If B is a singular matrix then…

A: Let A and B be two matrices of size 4 x 4

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. %3D If B is a singular matrix then…

A:

Q: Given A = [010; 0 0 1; 0 0 0], B= [0; 0; 1J, Calculate the rank of the Controllability matrix

A:

Q: If o is a normalised fundamental matrix of X = AX at to = 0 and f is continuous in t, then Xp(t) = |…

A:

Q: The Hessian matrix for f(x) = x³x3 + x2x3 at the point [1 2 3] is given by: [6 0 31 [18 0 31 [3 0 6]…

A:

Q: 3- If x=[ 26 12; 15 6 3; 10 11 1], then a) replace the first row elements of matrix x with its…

A: X=[  2 6 12;15 6 3;10 11 1]

Q: 4. Suppose that xa is an approximate solution to the system of Ax = b. Let A, and Tare nonsingular…

A: Sol

Q: Q of similarit Y between The rwationship Su9are matrices is

A: This is the problem of Linear Transfprmation.

Q: (g) Suppose A and B are real normal matrices satisfying AB* = B*A. Show that A+ B and AB are both…

A:

Q: A population with three age classes has a Leslie matrix L. If the initial population vector x0,…

A:

Q: Let A = 3 0 -1 2 Compute the indicated matrix. (If this is not possible, enter DNE in any single…

A: Solve  for the A^3

Q: 1 the Vector space P, of polynomials of degree at most two, the third column of the matrix of…

A: The given basis are: B= e1=4-x2,  e2=-2+3x2,  e3=1+x A= a1=1,  a2=x,  a3=x2

Q: 7. Find the Jacobian matrix of the following. a) x= (u+v),y = (u-v).= = 2uvw b) x=r cos 0, y =r sin…

A:

Q: 3) If x=[2 6 12; 15 6 3; 10 11 11, then: a) Replace the first row elements of matrix x with its…

A:

Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) O Non of all the…

A:

Q: et A be the following matrix 2 7 2 4 4 8 2 3 3 fter diagonalization, Pand Dare found such that A =…

A:

Q: If A is an n×n matrix having linearly independent eigenvectors v1,v2, . . . ,vn, then the n×n matrix…

A: Matrix and whose columns which are linearly independent

Q: (b) Compute the covariance matrix of the data given by x1 = [2 1]', x2 = [3 %3D x3 = [2 3]'and x4 =…

A: To answer the given question we shall use the following formula :  Let X and Y be two random…

Q: Let X and Y be jointly Gaussian random variables where oy= oy and p =-1. Find a transformation…

A: